Right-angled billiards and volumes of moduli spaces of quadratic differentials on ${\mathbb {C}}\!\operatorname{P}^1$

Abstract

We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on CP 1 and counting functions of the number of (bands of) simple closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago.Applying ergodic techniques to the Teichmüller geodesic flow we obtain quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards.Contents 1. Introduction 1 2. Configurations and Counting Theorems 12 3. Billiards in right-angled polygons and quadratic differentials 17 4. Values of the Siegel-Veech constants 21 5. Computation of the volumes of the moduli spaces 39 6.Counting trajectories and ergodic theory on moduli space 40 A. Proof of combinatorial identity 42 B. Counting pillowcase covers 54 C.